NASA-TM-111522

EFFECTS OF TANGENTIAL EDGE CONSTRAINTSON THE POSTBUCKLING BEHAVIOR OF FLAT ANDCURVED PANELS SUBJECTED TO THERMAL AND

MECHANICAL LOADS

by

L. Librescu and W. Lin

Virginia Polytechnic Institute and State UniversityBlacksburg, Virginia

and

M. P. Nemeth and J. H. Starnes, Jr.

NASA Langley Research CenterHampton, Virginia

Presented at the Symposium on Buckling andPostbuckling of Composite Structures

ASME Winter Meeting

November 6-11, 1994Chicago, Illinois

https://ntrs.nasa.gov/search.jsp?R=19960023884 2018-05-22T05:38:08+00:00Z

EFFECTS OF TANGENTIAL EDGE CONSTRAINTS ON THEPOSTBUCKLING BEHAVIOR OF FLAT AND CURVED PANELS

SUBJECTED TO THERMAL AND MECHANICAL LOADS

L. Librescu and W. Lin

Virginia Polytechnic Institute and State UniversityBlacksburg, Virginia

M. P. Nemeth and J. H. Starnes, Jr.

NASA Langley Research CenterHampton, Virginia

ABSTRACT

A parametric study of the effects of tangential edge constraints on the postbuckling response offlat and shallow curved panels subjected to thermal and mechanical loads is presented. Themechanical loads investigated are uniform compressive edge loads and transverse lateral pressure.The temperature fields considered are associated with spatially nonuniform heating over thepanels, and a linear through-the-thickness temperature gradient. The structural model is based ona higher-order transverse-shear-deformation theory of shallow shells that incorporates the effectsof geometric nonlinearites, initial geometric imperfections, and tangential edge motionconstraints. Results are presented for three-layer sandwich panels made from transverselyisotropic materials. Simply supported panels are considered in which the tangential motion ofthe unloaded edges is either unrestrained, partially restrained, or fully restrained. These resultsfocus on the effects of the tangential edge restraint on the postbuckling response. The results ofthis study indicate that tangentially restraining the edges of a curved panel can make the panelinsensitive to initial geometric imperfections in some cases.

INTRODUCTION

Major portions of the structure of high-speed aerospace vehicles consist of fiat and curvedpanels that are used as primary load carrying components. Some typical applications of thesepanels are aircraft stabilizers and fuselage sections and missile nose and body sections. Often,the preliminary design of these panels is based on the simplification that the edges are simplysupported and that the supports are completely rigid. In practice, however, the supports are notcompletely rigid and they can deform elastically, especially in the tangential directionperpendicular to the panel edges. A tangential direction, as used herein, is intended to mean thedirection of the unit outward normal to a panel edge in the plane tangent to the panel surface at

any point on the panel edge. Thus, understanding the effects of tangential edge constraints onthe postbuckling behavior of fiat and curved panels is an important consideration in the design ofthese panels.

Thepresentpaperaddressesthe effects of tangential edge restraints on the postbucklingbehavior of three-layer fiat and shallow curved sandwich panels made of transversely isotropicmaterials and subjected to combined thermal and mechanical loads. The panels have uniformthickness and are balanced and symmetric laminates. The mechanical loads include both lateral

pressure and uniform edge compression loads and the thermal loads include both spatiallynonuniform temperature fields and linear through-the-thickness temperature gradients. A specialpurpose analysis that is well-suited for parametric studies is described that includes the effects of

initial geometric imperfections and transverse-shear deformation. Results obtained using theanalysis are presented for simply supported panels that show the effects of thermal loads, lateralpressure, and edge compression loads on their postbuckling characteristics for the full range oftangential edge restraints including elastic edge restraints.

ANALYSIS DESCRIPTION

The analysis used in the present study is based on a higher-order transverse-shear-deformationtheory (HSDT) that includes the effects of geometric nonlinearities and initial geometricimperfections. The details of the theory are lengthy and only a summary of the analysis andsome of the details concerning the tangential edge constraints are presented in the present paper.Details of the analysis are given in Ref. 1.

Thermoelastic Constitutive Relations

The thermoelastic constitutive equations used for the present study are for symmetricallylaminated shallow curved panels with uniform thickness. The material is assumed to havethermoelastic transversely isotropic properties, with the plane of isotropy coinciding with thetangent plane at each point of the shell reference surface. The transversely isotropic constitutiveequations are characterized by five elastic constants and two thermal coefficients. The elasticconstants and thermal coefficients that characterize the plane of isotropy of the material are theelastic modulus E, Poisson's ratio v, and thermal compliance _. In addition, the elasticmodulus E', Poisson's ratio v', shear modulus G', and thermal compliance _' characterizethe material behavior perpendicular to the plane of isotropy. The coefficients of thermal

expansion a and ct' are related to the thermal compliances Z, and Z,' by equations presentedin Refs. 1 and 2. For transversely isotropic materials the transverse thermal expansioncoefficient tx' is often much larger than the tangential coefficient ct. In addition, thetransformed reduced thermoelastic constitutive equations used in the higher-order transverse-shear-deformation theory indicate that the thermal compliances depend on the ratio of the elasticmoduli E/E'. The corresponding equations based on classical shell theory do not have thisdependence. Another important characteristic of transversely isotropic materials is that thetransformed reduced thermal compliances can have negative values for ordinary values of E/E'and Z,/Z,' (see Refs. 1 and 2). These thermal compliances are always positive-valued for astrictly isotropic material.

Nonlinear Boundary-Value Problem

In the present study, the nonlinear equations governing the postbuckling response of shallowcurved panels are represented as an extension of the classical von Karman-Marguerre-Mushtarinonlinear shallow shell equations that include the effects of geometric imperfections andtransverse shear deformations. An Airy stress function is used to eliminate the shell inplaneforce equilibrium equations. Consequently, the compatibility equation for the membrane strainsis included as a primary field equation of the nonlinear boundary-value problem along with theremaining shell out-of-plane force equilibrium equation and the two moment equilibriumequations. A partially inverted form of the constitutive equations is introduced in which themembrane strains are expressed in terms of the Airy stress function and the transversedisplacement, and the bending stress resultants and transverse shear stress resultants are expressed

in terms of the rotations and the transverse displacement. Substituting these special constitutiveequations into the three remaining shell equilibrium equations and into the strain compatibilityequation yields four coupled partial differential equations in terms of the stress function, thetransverse displacement, and the two rotations. The equations are reduced further by expressing

the rotations in terms of the transverse displacement and a potential function F(xj,x2).

Substituting the resulting expression into the out-of-plane force equilibrium equation yields an

equation in terms of the stress function and transverse displacement, and substituting thisequation into the moment equilibrium equations yields a single Heimholz-type boundary layerequation in F that is totally uncoupled from the other equations. In general, however, theboundary-value problem remains coupled through the five boundary conditions at each edge ofthe shell.

The boundary conditions considered in the present study are simply supported boundaryconditions with varying degrees of tangential edge restraint. The tangential edge restraints act inthe direction of the unit outward normal to the panel edge in the plane tangent to the panel ateach point on a panel edge. For these boundary conditions, the transverse displacement at eachedge, the bending stress resultant acting about the axis parallel to each edge, and the rotationabout the axis normal to each edge (in the tangent plane) are all zero-valued. The degree oftangential edge restraint considered herein is bounded by the cases in which the tangential motionof the unloaded edges of a panel are either unrestrained or completely restrained, respectively, inthe inplane direction perpendicular to the panel edge. For these two cases, the panel edges arereferred to herein as moveable and immovable edges, respectively. All intermediate cases arereferred to herein as partially moveable edges and include elastically restrained edge constraints.

For a moveable edge, zero-valued tangential stress resultants are specified at the edge. Incontrast, for an immovable edge, the components of the tangential motion in the tangent planethat are normal and parallel to the edge are restrained and unrestrained, respectively. For thiscase, the shear stress resultant on the edge is specified as zero-valued, and the normaldisplacement in the tangent plane is specified as zero-valued in an average sense. To make thisdisplacement zero-valued in an average sense, the normal displacement in the tangent plane isobtained in terms of the transverse displacement and stress function by using the correspondingstrain-displacement relation and constitutive equations and then by integrating the resultingexpression over the planform of the shell. Setting the resulting equation equal to zero yields afictitious tangential stress resultant normal to the edge that makes the correspondingdisplacement zero-valued in an average sense.

The analytical procedure used for panels with opposite edges that are partially moveable issimilar to the procedure previously described for panels with immovable edges. In particular, the

average end-shortening displacement A_ between edges xl = 0 and x I = LI (see Fig. 1 for the

definition of the coordinate directions) is related to the corresponding compressive edge load Nil

byAic I =-Nil (la)

where c I is the average tangential stiffness in the x1 - direction on each opposite plate edge.

Similarly, for the edges x 2 = 0 and x2 = L 2

A2C 2 = - N22 (lb)

where c 2 is the average tangential stiffness in the x2- direction on each opposite plate edge.

The expressions for the end-shortening displacements are given by

-1AI = L---_ fo LI foL2 Ul,I dx2dXl (2a)

-1 f LlA2 = L I L 2 do f0 L2 u2'2 dx2dXl (2b)

where u I and u2 are the displacements in the x I and x2 directions, respectively, and a comma

followed by a subscript denotes partial differentiation. The average compressive edge loads aregiven by

I f0 L2 F22 Xl= dx2 (3a)-Nil _22 =O,Lj

lyon,I- = dxl (3b)N22 _ FII x2f0, L2

where F is the stress function (see Ref. 1) given by

F(xl,x2) = Fl(xl,x2)- _---[(x2)2 NIl+ (Xl) 2 N22] (4)

The partially restrained edge conditions are obtained by substituting Eqs. (2), (3), and (4) into

Eq. (1); expressing the displacements u I and u2 in terms of the transverse displacement and

stress function by using the corresponding strain-displacement relation and constitutive

equations; and then by solving for fictitious edges loads Nil and N22 that yield the desired

tangential stiffness constraint on the edges.

Equations (1) indicate that values of Aj =0 and A2 =0 correspond to immovable edges at

x ! = 0 and L I and x2 = 0 and L2, respectively. These conditions are enforced by selecting

c I = o_ and c 2 =*o, respectively. In addition, values of c I = 0 and c 2 = 0 correspond to

moveable edges at x I = 0 and L I and x2 = 0 and L2, respectively. For these moveable edge

conditions, NI! =0 and N22 = 0.

To measure the degree of edge restraint in a convenient way, alternate tangential stiffness

parameters _,! and L2 are introduced herein such that _,l = 0 and _,! = 1 correspond to

moveable and immovable edges at x I = 0 and Lt, respectively. Similarly, L 2 = 0 and k 2 = 1

correspond to moveable and immovable edges at x 2 =0 and L2, respectively. Partially

restrained edges at x I =0 and L! and x2=0 and L 2 aredefined by 0<L! < 1 and 0<L2< l,

respectively. These alternate tangential stiffness parameters are given by

(8 + _) c!_! - (5a)

! +(fi+_)cl

(b + _) c2_2 - (5b)

1 +(b+_)c 2

where (b + E) is the tangential stiffness quantity defined in Ref. 1.

Forthesimplysupportedboundaryconditionsdescribedabove,thenonlinearboundary-valueproblemis uncoupledwith respectto thepotentialfunction F andthesolutionto theHelmholz-typeboundarylayerequationin F is F = 0. Thus, for shells with the simplysupported boundary conditions described herein, the nonlinear boundary-value problem reduces totwo partial differential equations in terms of the Airy stress function and the transverse deflection

v3. These two equations are referred to herein as the von Karman-type compatibility and

transverse force equilibrium equations.

Solution of the Nonlinear EQuations

The nonlinear boundary-value problem in the present study is solved using Galerkin's method.

First, the transverse deflection v3 is expressed in terms of functions that satisfy the simply

supported boundary conditions

V3(XI,X2) = Wren sin_mxl sinlanX2 (6a)

where _m = mn/L1, _t, = nl_/L2, and Wren are the modal amplitudes and L l and L 2 are the

panel side lengths. Following the results presented in Ref. 3, the initial geometric imperfection

03 is expressed as

_3(Xl,X2) = _/tnn sin2_mxl sinlanx2 (6b)

where ffmn are the modal amplitudes of the initial geometric imperfection shape. Similarly,

the applied temperature and pressure fields are most generally represented by Navier-type doubleFourier sine series. In the present study, the temperature and pressure fields are approximated by

_(xl,x2) = "_mnsinXmxl sin_.x2

qF(xl,x2) = _m. sin_mX, sin_nX2

P3(Xl,X2) =Pm sinkmXl sinlx.x2

(6c)

(6d)

(6e)

where _ and "t?enter the analysis through the constitutive equations and are expressed in terms

of the temperatures at x3 = h/2 and -hi2 (these temperatures are denoted by Tu and Tb,

respectively) and are given by

=1 :(T u+ 'I'b) (7a)

+=l g(T.-Tb) (7b)

The temperature field is given by

T(Xl,X2,X3) = '_(Xl,X,:, ) + X3 '_(Xl,X2) (7c)

The displacement expansions are substituted into the von Karman-type compatibility equationand the Airy stress function is obtained by solving the resulting linear nonhomogeneous partialdifferential equation. The remaining nonlinear partial differential equation is the von Karman-type equilibrium equation and is converted into a set of nonlinear algebraic equations using

Galerkin'smethod.Thisprocedure yields the following set of M x N nonlinear algebraicequations for each set of wave forms determined by the index pair (m,n)

Rrs wrs +Pn Brs - H _'rs Crs

d +P3[w , o]+ 0 (8)

where the indices r and s are not summed and have the values r = 1, 2 ..... M and s = 1,

2 ..... N. In Eqs. (8), PI and P4, P2, and P3 are linear, quadratic, and cubic polynomials of the

unknown modal amplitudes wrs, respectively. The coefficients Brs, Crs, and Rrs are

constants that depend on the material and geometric properties of the shell and _11 and r_22 are

normalized forms of the tangential stress resultants representing the mechanical loads.The equilibrium configurations for a given flat or curved panel are obtained by solving the

nonlinear algebraic equations given by Eqs. (8) by using Newton's method. Every solution toEqs. (8) represents a possible stable or unstable equilibrium configuration. The stability of eachequilibrium configuration is determined by evaluating the second variation of the total potential

energy of the panel. Expressing Eqs. (8) symbolically as L(wrs) = 0 the second variation of

the total potential energy of the panel is obtained by computing the Jacobian of L with respect

to wr_ and evaluating it for each equilibrium configuration that is a solution of Eqs. (8). The

stable equilibrium configurations are given by those that make the Jacobian positive definite.

RESULTS AND DISCUSSION

The results presented in the present paper focus on the effects of tangential edge restraints onthe postbuckling behavior of simply supported flat and curved panels. The panels considered

have a square planform with dimensions L_ = L: = _ and a uniform thickness h. Results for flat

panels are presented in Figs. 1 through 4, and results for curved panels are presented in Figs. 5through 13. In these figures, results are presented for varying degrees of tangential edge restrainton opposite edges including the limiting cases of moveable and immovable edges. The degree of

tangential edge restraint is indicated in the figures by the values of the parameters _,l for the

edges x i = 0 and _, and by _'2 for the edges x2 = 0 and Q. A value of zero for either of these

parameters corresponds to moveable edges and a value of one corresponds to immovable edges.

Partially restrained edges at x I =0 and _ and x2=0 and _ are defined in the figures by

0 ¢_ _'1 ¢_ | and 0 < _'2 < 1, respectively. The thick solid and dashed lines shown in the figures

corresponds to the limiting cases of moveable and immovable edges, respectively.The panels considered herein consist of three layers of elastic transversely isotropic material.

For all of the results presented herein, values of v = 0.2 and a = 1.15 x 10 -6 in/in/°F were

used for Poisson's ratio, and the coefficient of thermal expansion of each layer, respectively. Inaddition, a value of v' = 0.2 was used for the through-the-thickness Poisson's ratio. The facesheets of each sandwich panel are identical, and the core is twice as thick as a face sheet.Moreover, the elastic moduli and thermal compliance coefficients for the face sheets are constantnondimensional ratios with values of (E'/E)f = 5, (E/G')f = 10, and (_,/_,')f = 1.4286,

respectively. Similarly, the elastic moduli and thermal compliance coefficients for the core areconstant nondimensional ratios with values of (E'/E) c = 2, (E/G')c = 30, and (_/_,')c = 1.2143,respectively.

Theresultsarepresentedin the form of curves that relate the magnitude of the average middle

surface temperature "_" at the center of a panel to the central transverse deflection or the average

end shortening as a function of the degree of tangential edge restraint, and in some cases asubcritical compressive edge load or transverse pressure load. The applied subcritical inplane

compression loads Nzl and N22, and the applied transverse pressure load P3 are represented

in the figures by the following nondimensional parameters

LII - Nil/r4 D (9a)

_22 = N22 _____22n 4 D (9b)

_

p3(0/2,_/2 ) _4

Dh (10)

where D is the isotropic bending stiffness of the panel given by

Eh 3D= (ll)

12(1 -v 2)

At buckling, the parameters ['_ll and _22 correspond to the well-known definitions of the

bifurcation buckling coefficients for isotropic panels that are denoted herein by (_ll)cr and

(f"22)cr" For example, for the square flat panels with moveable simply supported edges

considered herein, the buckling coefficient has a value of 4 for uniaxial compressive loading.Similarly, nondimensional central transverse deflection 8 and corresponding imperfectionamplitude 8o are used in the figures and are given by

v3(_2,_2)8=-- (12)

h

_3(_2,_2)80 --- (13)

h

For all the results presented herein, the temperature field is specified to be sinusoidallydistributed across each surface of the panels considered in this study (see Eq. (6)). Two through-the-thickness temperature distributions are considered in this study. One temperature distribution

has the same temperature (T u = T b) on both panel surfaces, and the other temperature distribution

has unequal temperatures (T u;e T b ) on the panel surfaces. For all results presented involving a

through-the-thickness temperature gradient, a constant value of Tb -- 70°F was used to obtain

the results. Likewise, a value of the panel length-to-thickness ratio _/h = 100 was used toobtain the results.

Studies of the effects of uniform temperature increase (T u = Tb) on several of the postbuckling

equilibrium configurations of a typical fiat panel considered herein were conducted. These results

indicate that the stable postbuckling equilibrium configurations correspond to shapes given bym = n -- 1 in Eq. (6a). This trend was found to be true for the range of structural parametersconsidered herein. In all subsequent figures, the magnitude of the temperature distribution given

by "_lt is denoted by "_ in the figures for convenience.

Results for Flat Panels

The effects of a tangential edge restraint on the postbuckling behavior of flat panels subjected

to a uniform through-the-thickness temperature increase (T, = Tb) are show in Figs. 1 through

3. The results shown in Figs. 1 and 2 are for panels with moveable edges at xj = 0 and _ and

with varying degrees of equal tangential edge restraint at x 2 = 0 and _. The results shown in

Fig. 3 are for panels with both pairs of opposite edges having varying degrees of equal tangentialedge restraint. Each of the panels shown in these figures have perfectly flat prebuckling shapes,and buckle due to the spatially nonuniform heating applied to the panel top and bottom surfaces.Buckling occurs at the bifurcation points indicated on the figures by the filled circles.

The results presented in Figs. 1 and 2 for panels with one pair of opposite edges equallyrestrained indicate that the buckling temperature decreases as the degree of tangential edgerestraint increases. The lowest and highest buckling temperatures are for the panels withimmovable and moveable edges, respectively. This trend is due to the increase in compressivestress that results from increasing the tangential edge restraint. Comparing the results presentedin Figs. 1 and 3 shows that this effect is amplified when both pairs of opposite edges arerestrained. The results in Figs. I through 3 also show that the postbuckling stiffness increases

in the deep postbuckling range (8 > 2.0) as the degree of tangential edge restraint increases.

This behavior is attributed to an increase in panel membrane stiffness as the degree of tangentialedge restraint increases.

The effects of tangential edge restraint on the postbuckling behavior of flat panels subjected to

a linear through-the-thickness temperature increase (T u _ Tb ) are show in Fig. 4. The results in

this figure are for panels with both pairs of opposite edges having varying degrees of equaltangential edge restraint. These panels begin deflecting out-of-plane at the onset of applying thethermal load due to the bending moments induced by the through-the-thickness thermal gradientand do not have a bifurcation buckling load. Comparing these results with the results in Fig. 3shows that the effect of a tangential edge restraint is much smaller for panels subjected to athrough-the-thickness temperature gradient than for corresponding panels subjected to a uniformtemperature. In addition, the results show an increase in overall bending stiffness with anincreasing degree of tangential edge restraint which is more pronounced in the initial stage of thepostbuckling response for panels with a through-the-thickness temperature gradient than forpanels with a uniform through-the-thickness temperature.

Results for Curved Panels

The effect of tangential edge restraints on the nonlinear behavior of cylindrical panels with

/R 2 = 0.3 and subjected to a uniform through-the-thickness temperature increase is shown in

Fig. 5. The results in this figure are for geometrically perfect cylindrical panels with tangentialedge restraints on the curved edges. The straight edges are moveable and loaded by a uniform

compressive edge load that is 75% of the corresponding bifurcation buckling load; i.e.,

_22 = 0.75 (["22)cr" These results indicate a monotonic temperature-deflection behavior for the

panels for the full range of tangential edge restraint. This monotonic behavior is a result of thedeflections associated with the uniform heating of a curved panel that increase its curvature in astable manner (i.e., the panel deflects away from its projected planform when it is heated). In

addition,thepanelswith immovableedgesareshownto havelargerdeflectionsthanthecorrespondingpanelswithmoveableedgesatagivenvalueofthetemperature.Thisbehaviorisaresultof additionaltransversedeflectionsduetotheadditionalcompressionloadinducedbyrestrainingthecurvededgesofthepanels.Theeffectsofatangentialedgerestraintonthenonlinearbehaviorofveryshallowcylindrical

panels(_/R2= 0.03)subjectedtoathrough-the-thicknesstemperaturegradientareshowninFig.6. Theresultsinthisfigureareforgeometricallyperfectcylindricalpanelswiththecurvededgestangentiallyrestrained.Thestraightedgesaremoveableandloadedby auniformcompressiveedgeload_22=0.75(I_22)cr. TheresultsinFig.6 showthattheresponsecurvesforthesepanelshavealimitpointforvaluesofthetangentialedgerestraintparameter_,1< 0.3,

and that the thermal limit load increases as the value of _,t increases. For _,1 = 0.34, the panel

response curve has a bifurcation point. For _,_> 0.34, the panel response curve is characteristic

of a stable monotonic temperature-deflection response.The effects of initial geometric imperfections on the response of the curved panels represented

in Fig. 6 are shown in Figs. 7 and 8. The results in Figs. 7 and 8 are for curved panels withpositive initial geometric imperfections with amplitudes of 0.1 and 0.2, respectively. Apositive imperfection amplitude is inward. The panels with the smaller imperfection amplitudehave similar behavioral trends as the corresponding perfect panels shown in Fig. 6. However,the results for the imperfect panels indicate that the effects of the tangential edge restraint aremore pronounced than for the perfect panels and that there is a substantial reduction in the

thermal limit load for a given value of kl- The limit points for the imperfect panels occur for

much higher values of _,1 than for the perfect panels. The imperfect panels shown in Fig. 7

have a bifurcation point for kl = 0.67, and a monotonic nonlinear behavior for values of

_,_ > 0.67. These differences in behavior indicate that tangentially restraining the edges of a

curved panel can make the panel insensitive to initial geometric imperfections in some cases.The results in Figs. 7 and 8 indicate that increasing the geometric imperfection amplitude of

the panels eliminates the bifurcation and monotonic nonlinear responses of the panels. Panels

with the larger imperfection amplitude 5o = 0.2 have a limit point behavior for the most part and

panels with immovable curved edges have the highest value of thermal limit load.The effects of subcritical circumferential compressive edge loads on the response of the curved

panels represented in Fig. 6 are shown in Figs. 9 and lO. These suberitical edge loads for Figs.

9 and 10 have values given by L,22= 0.5 (]['22)cr and I_22 = 0, respectively. These results, and

the results in Fig. 6, show that the response trends are very sensitive to the presence andmagnitude of the circumferential compressive edge loads. The results show that reducing the

value of I-,22 significantly reduces the sensitivity of the response to the degree of tangential

edge restraint except in the deep postbuckling range of behavior. Similarly, reducing the value

of _22 significantly changes the response trends of the panels. Panels with L22 = 0.5 (]['22)cr

have a limit point response for the most part. The panels with the immovable edges have the

highest value of thermal limit load. In contrast, all panels with ][,22= 0 have monotonically

increasing transverse deflections with increasing temperature.The effects of a tangential edge restraint on the nonlinear behavior of shallow cylindrical panels

(_ / R 2 = 0.1) subjected to a through-the-thickness temperature gradient are shown in Figs. 1 1

and 12. The results in these figure are for geometrically perfect cylindrical panels with theirstraight edges tangentially restrained. The curved edges are moveable and loaded by a uniform

compressive edge load given by L,li =0.75 (L'll)er" These results, and the results in Fig. 6

suggestthatpanelswithrestrainedstraightedgesandloadedcurvededgesgenerallyhavethesameresponsetrendsaspanelswithrestrainedcurvededgesandloadedstraightedges.ThepanelsofFigs.11and12havealimitpointresponseforvaluesofthetangentialedgerestraintparameterL 2 _<0.1. The highest value of the thermal limit load for these panels is for L2 = 0.1. For

L 2 =0.18, the panels have a bifurcation buckling load and for k 2 >0.18, the panels have a

monotonic temperature-deflection response.

The effects of tangential edge restraints on the nonlinear behavior of shallow cylindrical panels

(Q/R 2 = 0.1) subjected to a through-the-thickness temperature gradient are shown in Fig. 13.

The results in the figure are for geometrically perfect panels with the straight edges tangentiallyrestrained and with moveable curved edges. The panels are loaded by an external transverse

pressure load P3, given by Eq. (6e) with m = n = 1. The magnitude of the pressure load isgiven by p = 2 (see Eqn. 10). The results shown in Figs. 6, 11, and 13 indicate that the effectsof the degree of tangential edge restraint on the nonlinear behavior is qualitatively the same forpanels that are loaded by transverse pressure and by inplane compressive loads. Comparing the

results in Figs. 11 and 13 indicates that the pressure-loaded panels appear to be slightly moresensitive to the degree of tangential edge restraint than the compression-loaded panels. For

example, the pressure-loaded panel with moveable straight edges (_ = 0) has monotonically

increasing transverse deflections with no limit point with increasing temperature and thecorresponding panel with compression loaded curved edges has a limit point. In addition, the

pressure-loaded panels have a bifurcation point for L 2 = 0.8 and the corresponding compression-

loaded panels shown in Fig. 11 have a bifurcation point for k2 = 0.18.

CONCLUDING REMARKS

The results of a parametric study of the postbuckling behavior of flat and curved panelssubjected to thermal and mechanical loads have been presented. The mechanical loads consideredin the study are uniform compressive edge loads that are less than the panel buckling loads andtransverse lateral pressure. The temperature fields considered are associated with spatiallynonuniform heating over the panel surfaces and a linear through-the-thickness temperaturegradient. The analytical model used in the study is based on a higher-order transverse-shear-

deformation theory of shallow shells that incorporates the effects of geometric nonlinearites andinitial geometric imperfections. Results are presented for simply supported three-layer sandwichpanels made from a transversely isotropic material.

The results presented herein show that flat panels subjected to a uniform through-the-thicknesstemperature are generally more sensitive to the degree of tangential edge restraint than thecorresponding panels subjected to a through-the-thickness temperature gradient. For thecylindrical panels considered herein, the results show that the degree of tangential edge restraintcan significantly change the character of the nonlinear response and, in some cases, make a panelinsensitive to initial geometric imperfections. The results also show that the effects of atangential edge restraint are more pronounced for geometrically imperfect curved panels than forthe corresponding perfect panels, and that the response is more sensitive to the degree oftangential edge restraint when compressive edge loads are applied.

ACKNOWLEDGMENT

The work reported herein was partially supported by NASA Grant NAG 1-1300.

10

REFERENCES

. Librescu, L., Lin, W., Nemeth, M. P., and Starnes, J. H., Jr., "Classical Versus Non-

Classical Postbuckling Behavior of Laminated Composite Panels Under Complex LoadingConditions," AMD-Vol. 164, Non-Classical Problems of the Theory and Behavior ofStructures Exposed to Complex Environmental Conditions, ASME, 1993, pp. 169-182.

2. Librescu, L., "Elasto-Statics and Kinetics of Anisotropic and Heterogene90_ Shell-Type

Structures," Noordhoff International Publishers, The Netherlands, Leyden, 1975.

. Seide, P., "A Reexamination of Koiter's Theory of Initial Postbuckling Behavior andImperfection Sensitivity of Structures," Thin Shell Structures: Theory. Experiment and

D..c.Ei.ga," Y. C. Fung and E. E. Sechler, Eds., Prentice-Hall, 1974, pp. 59-80.

Temperature,

"_'(OF)

2000F J• Bifurcation point1600

1200 =7

8OO

\ 0 < Z.2< 1 _ TU400

_,2= 1

I

0.6

T u = T b

0 I I | I0 1.2 1.8 2.4 3

Nondimensional transverse deflection, _5

Fig. 1 Effects of tangentially restraining the panel edges at x2 = 0 and x2 = _ on the

postbuckling response of flat panels (edges x| = 0 and xj = _ are moveable)

subjected to uniform heating.

11

2OOO

1600

Temperature,

"I'(OF) 12oo

800

400

0

• Bifurcation point V x

"/ L

0 < ;_2< 1 T = T b

, _2,= 1[ ,_ , , , , ,

-0.0004 -0.0002 0 0.0002 0.0004

Nondimensional end-sho_ening, A

Fig. 2 Effects of tangentially restraining the panel edges at x2 = 0 and x2 = Q on the

postbuckling response of flat panels (edges x] = 0 and x I = _ arc moveable)

subjected to uniform heating.

0001• Bifurcation point1600

X 1Temperature,

_'(°F) 1200 _'1 ---- _'2" 0 (_ 7'

800

400_ 0 < _'1" _'2 < 1... .. "-

0 0.6 1.2 1.8 2.4 3

Nondimensional transverse deflection, 6

Fig. 3 Effects of tangentially restraining all panel edges on the postbuckling responseof flat panels subjected to uniform heating.

12

1000

800

Temperature,

1"(°F) so0

400 _'1 = _2 = 0

0 < ;k1= ;k2< 1

200

_1 = _2 = 1

0 I I I I I0 0.4 0.8 1.2 1.6 2

Nondimensional transverse deflection, 6

Fig. 4 Effects of tangentially restraining all panel edges on the postbuckling responseof flat panels subjected to a through-the-thickness temperature gradient.

2000 I

Temperature:e°° [

-2

=0

T u = T b

I

-1.7 -1.4 -1.1 -0.8

Nondimensional transverse deflection, 6

-0.5

Fig. 5 Effects of tangentially restraining the curved edges on postbuckling response ofcylindrical panels subjected to uniform heating and compressive edge loads

( /R2 = 0.3, ,22=0.75

13

04_-,_T_----x_ _.3

1200[ \LO1000 .34

Temperature, 00o 0.3

I"(°F) 0.2 R 2 0.2

0oo 0.1 0.1

400 0

/ ./ W -1 - _ _1---- 0.34

2oo1" _,1= 1 I _ _'• Bifurcation point

0-2 -1 0 1 2

Nondimensional transverse deflection,

Fig. 6 Effects of tangentially restraining the curved edges on the postbuckling response ofgeometrically perfect cylindrical panels subjected to a through-the-thickness temperature

gradient and a compressive edge load (_/R 2 = 0.03, L,22= 0.75 (_22)cr).

lOOO[ '_"_,_/l'' 0.7 .,__(__,x"(_ _=0.1o

00o_ ",_\_ I N22_ ,#_ 0.67Temperature, J "_\\\J 0.67

600 ""I"(°F) _---,_... :i:.._7_--- x,

_'1=4OO

,oot Wfo o• Bifurcation point

0-1.5 -1 -0.5 0 0.5 1 1.5 2

Nondimensional transverse deflection plus imperfection,8+_i

o

Fig. 7 Effects of tangentially restraining the curved edges on the postbuckling response ofimperfect cylindrical panels subjected to a through-the-thickness temperature gradient and

a compressive edge load (_ / R2 = 0.03, L22 = 0.75 (['22)cr , _o = 0.1 ).

14

1000

_5° = 0.2 Xl

800 0.4 ,"

Temperature,

l"(°F) eoo

_'1= 1 u R2

. .0.7 ,J_"400

200

IW_ I I I I

-' -0.5 0 0.5 1 1.5 2

Nondimensional transverse deflection plus imperfection,

_+_o

Fig. 8 Effects of tangentially restraining the curved edges on the postbuckling response ofimperfect cylindrical panels subjected to a through-the-thickness temperature gradient and

a compressive edge load (_ / R2 = 0.03, [`22= 0.75 ([`22)c_ , So = 0.2).

1000Xl e,"

Temperature, coo N2 2

T(oF) 600 x

400 ;_1= 1____1_

0.7_ \_1--'-- 1

200 f Z,I= 0• I I I I I

-1 -0.5 0 0.5 1 1.5 2

Nondimensional transverse deflection, _5

Fig. 9 Effects of tangentially restraining the curved edges on the postbuckling response ofgeometrically perfect cylindrical panels subjected to a through-the-thickness temperature

gradient and a compressive edge load ([ / R2 = 0.03, [`22 = 0.5 ([.22).).

15

1000

800

Temperature,

TI°F) 600

40O

2OO

0-0.5

0.4Xl 1

_'1---- 0"___0 < _,1 < 1

t1=1, i i , ,

1

0 0.5 1 1.5 2

Nondimensional transverse deflection, 8

Fig. 10 Effects of tangentially restraining the curved edges on the postbuckling response ofgeometrically perfect cylindrical panels subjected to a through-the-thickness

temperature gradient and a compressive edge load ([ / R2 = 0.03, _22 = 0).

Temperature,

T(°F)

1200 F 0.2 x 1

/

1111 f

600 I

400 [

201 [ , i i i• Bifurcation point

-1 0 1 2 3

Nondimensional transverse deflection, 8

Fig. 11 Effects of tangentially restraining the straight edges on the postbuckling response ofgeometrically perfect cylindrical panels subjected to a through-the-thickness

temperature gradient and a compressive edge load (_ / R2 = 0. I, I_]i = 0.75 (l_ll)c,).

16

X 1

Temperature, / I _._ R.

/ _\_-o18 / _o18

,o0i2°°1- 1, X2= 0 .......

| _ " • Bifurcation point _o / I1 I I I I I

0 0.0005 0.001 0.0015 0.002 0.0025

Nondimensional end-shortening, A1

Fig. 12 Effects of tangentially restraining the straight edges on the postbuckling response of

geometrically perfect cylindrical panels subjected to a through-the-thickness

temperature gradient and a compressive edge load (Q/R 2 = 0.1, f'-Jl = 0.75 (_l])_r)-

2000

1600

Temperature,

TI°F)1200

800

400

0..

.9

tt_ •

|

|

P = P3 _4 X 1 /

,u -x,y_/_i_.6 _.2= 0--.._

-o _,,_/ ,-Bifurcation point ,

1 2 3 4

Nondimensional deflection,

Fig. 13 Effects of tangentially restraining the straight edges on the postbuckling response of

geometrically perfect cylindrical panels subjected to a through-the-thickness

temperature gradient and a transverse pressure load (_ / R 2 -- 0.1, p -- 2).

17